Dynamic Epistemic Logic in Update Logic

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1 Dynamic Epistemic Logic in Update Logic Guillaume Aucher To cite this version: Guillaume Aucher. Dynamic Epistemic Logic in Update Logic. Journal of Logic and Computation, Oxford University Press (OUP), 2016, 26 (6), pp < /logcom/exw002>. <hal > HAL Id: hal Submitted on 8 Apr 2017 HAL is a multi-disciplinary open access archive for the deposit and dissemination of scientific research documents, whether they are published or not. The documents may come from teaching and research institutions in France or abroad, or from public or private research centers. L archive ouverte pluridisciplinaire HAL, est destinée au dépôt et à la diffusion de documents scientifiques de niveau recherche, publiés ou non, émanant des établissements d enseignement et de recherche français ou étrangers, des laboratoires publics ou privés.

2 Dynamic Epistemic Logic in Update Logic Guillaume Aucher University of Rennes 1 INRIA 263, Avenue du Général Leclerc Rennes Cedex, France guillaume.aucher@irisa.fr August 12, 2016 Abstract We show that Dynamic Epistemic Logic (DEL) is a substructural logic and that it is an extension of the update logic introduced in the companion article [12]. We identify axioms and inference rules that completely characterize the DEL product update and we provide a sequent calculus for DEL. Finally, we show that DEL with a finite number of atomic events is as expressive as epistemic logic. In parallel, we provide a sequent calculus for update logic which turns out to be a generalization of the non-associative Lambek calculus. 1 Introduction Dynamic Epistemic Logic (DEL) is an influential logical framework for reasoning about the dynamics of beliefs and knowledge, which has drawn the attention of a number of researchers ever since the seminal publication of Baltag & Al. [21]. A number of contributions have linked DEL to older and more established logical frameworks: it has been embedded into (automata) PDL [113, 107], it has been given an algebraic semantics [18, 19], and it has been related to epistemic temporal logic [105, 14] and the situation calculus [104, 112]. Despite these connections, DEL remains, arguably, a rather isolated logic in the vast realm of non-classical logics and modal logics. This is problematic if logic is to be viewed ultimately as a unified and unifying field and if we want to avoid that DEL goes on riding off madly in all directions (a metaphor used by van Benthem [101, 103] about logic in general). In this article we will show that DEL can be redefined naturally and meaningfully as a two-sorted substructural logic. The well-known semantics for substructural logics is based on a ternary relation introduced by Routley and Meyer for relevance logic in the 1970 s [88, 89, 90, 91]. However, the introduction of this ternary relation was originally motivated by technical reasons, and it turns out that providing a non-circular and conceptually grounded interpretation of this relation remains problematic [31]. As we shall see, the ternary semantics of DEL provides a conceptual foundation for Routley and Meyer s semantics. In fact, the dynamic interpretation induced by the DEL 1

3 framework turns out to be not only meaningful, but also consistent with the interpretations of this ternary relation proposed in the substructural literature. More specifically, we will show that DEL can be embedded within the substructural framework. The update logic obtained in the companion article [12] for the substructural framework will be made more specific by the addition of inference rules. This will yield a sequent calculus for DEL. Doing so, we will develop a basic correspondence theory for DEL and we will elicit a number of axioms and inference rules that play an important role in the DEL setting. The article is structured as follows. To make it self-contained, we reintroduce a number of results and definitions of the companion article [12]. In Section 2 we recall the core of DEL viewed from a semantic perspective. In Section 3 we briefly recall elementary notions of relevance and substructural logics and we observe that the ternary relation of relevance logic can be interpreted as an update, like the DEL product update. In Section 4 we proceed further to define a generalized substructural language based on this idea. In Section 5 we recall the cut-free display calculus for update logic defined in the companion article [12] and we introduce a sequent calculus for update logic. This sequent calculus turns out to be a generalization of the non associative Lambek calculus. In Section 6 we show that DEL can be embedded within our update logic by providing inference rules that completely characterize the DEL product update. This leads us to define a sequent calculus for DEL. Finally, in Section 7, we formally relate our DEL substructural operators to the dynamic inferences introduced in [97] and the DEL-sequents introduced in [8, 9]. We also show in this section that DEL with a finite number of atomic events is as expressive as epistemic logic. We conclude in Section 8. Note. Parts of Sections 2, 3.2 and 7 appear in [10] with some minor changes. 2 Dynamic Epistemic Logic Dynamic epistemic logic (DEL) is a relatively recent non-classical logic [21] which extends ordinary modal epistemic logic [52] by the inclusion of event models (called L A -models in this article) to describe actions, and a product update operator that defines how epistemic models are updated as the consequence of executing actions described through event models (see [20, 110, 103] for more details). Thus, the methodology of DEL is such that it splits the task of representing the agents beliefs and knowledge into three parts: first, one represents their beliefs about an initial situation; second, one represents their beliefs about an event taking place in this situation; third, one represents the way the agents update their beliefs about the situation after (or during) the occurrence of the event. Following this methodology, we also split the exposition of the DEL framework into three sections. Notation 1. In the rest of this article, P is a countable set of propositional letters called atomic facts which describe static situations, and G := {1,..., m} is a finite set of indices called agents. 2.1 Representation of the Initial Situation: L P -model First, we define the epistemic language L P. 2

4 Definition 1 (Language L P ). We define the language L P inductively by the following grammar in BNF, where p ranges over P and j over G: L P : ϕ ::= p ϕ (ϕ ϕ) (ϕ ϕ) j ϕ We will use the following abbreviation: j ϕ := j ϕ and ϕ ψ := ϕ ψ. To save parenthesis, we use the following ranking of binding strength:,, (i.e., binds stronger than which binds stronger than ). For example, j p q r p means ((( j p) q) r) p. We denote by L P 0 the propositional language, that is, the sublanguage of L P defined on the basis of the propositional connectives, and. A (pointed) L P model (M, w) represents how the actual world represented by w is perceived by the agents. Atomic facts are used to state properties of this actual world. Definition 2 (L P -model). A L P -model is a tuple M = (W, R 1,..., R m, I) where: W is a non-empty set of possible worlds, R j W W is an accessibility relation on W, for each j G, I : W 2 P is an interpretation assigning to each possible world a subset of P. A L P -frame is a L P -model without interpretation. We write w M for w W, and (M, w) is called a pointed L P -model (w often represents the actual world). If w, v W, we write wr j v or (M, w)r j (M, v) for (w, v) R j, and R j (w) denotes {v W wr j v}. We denote by E P the set of pointed L P -models and by E F the class of pointed L P -frames. Intuitively, wr j v means that in world w agent j considers that world v might correspond to the actual world. The epistemic language can be used to describe and state properties of L P - models. The formula j ϕ reads as agent j believes/knows ϕ. 1 Its truth conditions are defined in such a way that agent j believes ϕ holds in a possible world when ϕ holds in all the worlds agent j considers possible. Definition 3 (Satisfaction relation for E P L P ). We define the satisfaction relation E P L P as follows. Let M be a L P -model, w M and ϕ, ψ L P. The truth conditions for the atomic facts and the connectives,, and j are defined as follows: M, w p iff p I(w) M, w ψ iff it is not the case that M, w ψ M, w ϕ ψ iff M, w ϕ and M, w ψ M, w ϕ ψ iff M, w ϕ or M, w ψ M, w j ϕ iff for all v R j (w), we have that M, v ϕ A L-formula ϕ is satisfiable if there is (M, w) E P such that M, w ϕ. 1 The reading of jϕ depends on the properties of the accessibility relation: if R j is reflexive (and transitive), then jϕ typically reads agent j knows ϕ, but if R j is only serial (and transitive) then jϕ reads agent j believes ϕ. 3

5 r A, b B, w C A w : r A, b C, w B B r C, b B, w A C B A C r C, b A, w B A r B, b A, w C C r B, b C, w A B Figure 1: Cards Example Example 1. Assume that agents A, B and C play a card game with three cards: a white one, a red one and a blue one. Each of them has a single card but they do not know the cards of the other players. At each step of the game, some of the players show their/her/his card to another player or to both other players, either privately or publicly. We want to study and represent the dynamics of the agents beliefs in this game. The initial situation is represented by the pointed L P -model (M, w) of Figure 1. In this example, G := {A, B, C} and P := {r j, b j, w j j G} where r j stands for agent j has the red card, b j stands for agent j has the blue card and w j stands for agent j has the white card. The boxed possible world corresponds to the actual world. The propositional letters not mentioned in the possible worlds do not hold in these possible worlds. The accessibility relations are represented by arrows indexed by agents between possible worlds. Reflexive arrows are omitted in the figure, which means that for all worlds v M and all agents j G, v R j (v). In this model, we have for example the following statement: M, w (w B A w B ) C A w B. It states that player A does not know that player B has the white card and player C knows it. 2.2 Representation of the Event: L A -model The language L A below was introduced by Baltag & Al. [22]. We call the expressions p π atomic events. Definition 4 (Language L A ). We define the language L A inductively by the following grammar in BNF, where π ranges over L P and j over G: L A : χ ::= p π χ (χ χ) (χ χ) j χ We use the same abbreviations as in Definition 1. Like for L P, we use the following ranking of binding strength:,,. A pointed L A -model (A, e) represents how the actual event represented by e is perceived by the agents. Intuitively, f R A j (e) means that while the possible event represented by e is occurring, agent j considers possible that the possible event represented by f is actually occurring. 4

6 Definition 5 (L A model, [21]). A L A -model is a tuple A = (W A, R A 1,..., R A m, I A ) where: W A is a non-empty set of possible events, R A j W A W A is an accessibility relation on W A, for each j G, I A : W A L P is a precondition function assigning to each possible event a formula of L P. Let P be a subset of L P. A P complete L A model is a L A model which satisfies moreover the following condition: I A (e) P, for each e W A (P-complete) A L A frame is a L A model without precondition function I A. We abusively write e A for e W A, and (A, e) is called a pointed L A -model (e often represents the actual event). We abusively write I A (A, e) for I A (e). We denote by E A the set of pointed L A -models, by EP A the set of pointed P -complete event models and by EA F the class of pointed LA -frames. If e, f W A, we write erj A f or (A, e)rj A (A, f) for (e, f) Rj A, and Rj A (e) denotes {f W A erj A f}. The truth conditions of the language L A are almost identical to the truth conditions of the language L P : Definition 6 (Satisfaction relation for A L A ). We define the satisfaction relation A L P as follows. Let A be a L A -model, e A and χ, ρ L A. The truth conditions for the atomic events and the connectives,, and j are defined as follows: A, e p π iff I A (e) = π A, e χ iff it is not the case that A, e χ A, e χ ρ iff A, e χ and A, e ρ A, e χ ρ iff A, e χ or A, e ρ A, e j χ iff for all f Rj A (e), we have that A, f χ Remark 1. The current reading of p π is an event of precondition π is occurring. However, other truth conditions could be provided for the atomic events p π. We list some of them below: A, e p π iff I A (e) π (1) A, e p π iff I A (e) π is satisfiable (2) The truth condition of Expression (1) states that the situation where the event e occurs must necessarily satisfy the property π. The truth condition of Expression (2) turns out to be expressible in terms of Expression (1) ( p π ). Example 2. Let us resume Example 1 and assume that players A and B show their card to each other. As it turns out, C noticed that A showed her card to B but did not notice that B did so to A. Players A and B know this. This event is represented in the event model (A, e) of Figure 2. The boxed possible event e corresponds to the actual event players A and B show their red and white 5

7 A,B e : r A w B f : w A C C C g : r A A,B,C A,B,C Figure 2: Players A and B show their cards to each other in front of player C A,B,C e : r A Figure 3: Public announcement of r A cards respectively to each other (with precondition r A w B ), f stands for the event player A shows her white card (with precondition w A ) and g stands for the atomic event player A shows her red card (with precondition r A ). The following statement holds in the example of Figure 2: A, e p ra w B ( A p ra w B A p ra w B ) ( B p ra w B B p ra w B ) ( C p wa C p ra C (p wa p ra )) (3) It states that players A and B show their cards to each other, players A and B know this and consider it possible, while player C considers possible that player A shows her white card and also considers possible that player A shows her red card, since he does not know her card. In fact, that is all that player C considers possible since he believes that either player A shows her red card or her white card. 2 The L A -model of Figure 3 corresponds to a public announcement or public display of the fact that agent A has the red card. In particular, the following statement holds in the example of Figure 3: A, e p ra A p ra B p ra C p ra A A p ra A B p ra A C p ra B A p ra B B p ra B C p ra C A p ra C B p ra C C p ra... It states that player A shows her red card and that players A, B and C know it, that players A, B and C know that each of them know it, etc. in other words, there is common knowledge 2 Note that accessibility relations for C are no longer reflexive in (A, e), so we use the term belief instead of knowledge for player C. 6

8 r A, b C, w B A r A, b B, w C C A,B (w, e) : r A, b C, w B C C r B, b C, w A A r C, b B, w A Figure 4: Situation after the Update of the Situation Represented in Figure 1 by the Event Represented in Figure 2 among players A, B and C that player A shows her red card. 3 A, e p ra G p r A. 2.3 Update of the Initial Situation by the Event: Product Update The DEL product update of Baltag & Al. [21] is defined as follows. This update yields a new L P - model (M, w) (A, e) representing how the new situation which was previously represented by (M, w) is perceived by the agents after the occurrence of the event represented by (A, e). Definition 7 (Product update). Let (M, w) = (W, R 1,..., R m, I, w) be a pointed L P -model and let (A, e) = (W A, R1 A,..., Rm, A I, e) be a pointed L A -model such that M, w I A (e). The product update of (M, w) and (A, e) is the pointed L P -model (M A, (w, e)) = (W, R1,..., R m, I, (w, e)) defined as follows: for all v W and all f W A, W = {(v, f) W W A M, v I A (f)}, R j (v, f) = {(u, g) W u R j (v) and g R A j (f)}, I (v, f) = I(v). Example 3. As a result of the event described in Example 2, the agents update their beliefs. We get the situation represented in the L P -model (M, w) (A, e) of Figure 4. In this model, we have for example the following statement: (M, w) (A, e) (w B A w B ) C A w B. It states that player A knows that player B has the white card but player C believes that it is not the case. 3 We write A, e Gχ when for all f j GR j A (e), we have A, f χ. See for example Fagin & Al. [38] for a detailed study of the operator G of common knowledge. 7

9 Weakening: Contraction: Permutation: U V X, U V W A X, X, U V X, U V U, Y, X, V W C A U, X, Y, V W P A U U V V, X W K U V, X, X U V, X C K U U V, Y, X, W V, X, Y, W P K Figure 5: Gentzen s Structural Rules for Classical Logic 3 Updates as a Conceptual Foundation for Substructural Logics Substructural logics are a family of logics lacking some of the structural rules of classical logic. A structural rule is a rule of inference which is closed under substitution of formulas [86, Definition 2.23]. In a certain sense, a structural rule allows to manipulate the structure(s) of the sequent/consecution without altering its logical content. The structural rules for classical logic introduced by Gentzen [47] are given in Figure 5. The comma in these sequents has to be interpreted as a conjunction in an antecedent and as a disjunction in a consequent. While Weakening (W A, W K ) and Contraction (C A, C K ) are often dropped as in relevance logic and linear logic, the rule of Permutation (P A, P K ) is often preserved. When some of these rules are dropped, the comma ceases to behave as a conjunction (in the antecedent) or a disjunction (in the consequent). In that case the comma corresponds to other substructural connectives and we often introduce new punctuation marks which do not fulfill all these structural rules to deal with these new substructural connectives. 3.1 Substructural Logics Our exposition of substructural logics is based on [86, 87, 37] (see also [77] for a general introduction). 4 The logical framework presented in [86] is more general and studies a wide range of substructural logics: relevant logic, linear logic, Lambek calculus, arrow logic, etc. We will only introduce a fragment of this general framework in order to highlight the main new ideas. In particular, we will not consider truth sets and we will assume that our logics do not reject distribution. These other features can be added and our framework can be adapted, following the exposition of Restall [86]. We will moreover assume that we have multiple modalities (one for each agent j G). The semantics of substructural logics is based on the ternary relation of the frame semantics for relevant logic originally introduced by Routley and Meyer [88, 89, 90, 91]. Another semantics proposed independently by Urquhart [93, 94, 95] at about the same time will be discussed at the end of this section. 4 We very slightly change the definitions of frames and models as they are defined in [86] (we give the details of these differences in the sequel). The definitions remain equivalent nevertheless. 8

10 In the sequel we consider the following set of logical connectives: } Sub := { j, j,,,,,,,,, j G We also define the set of connectives Sub := Sub { } (the connective corresponds to the intuitionistic implication). Definition 8 (Languages L(P, Sub) and L(P, Sub )). The language L(P, Sub) is the language associated to Sub, that is, the language built compositionally from the connectives of Sub and the set of propositional letters P. More formally, it is the set of formulas defined inductively by the following grammar in BNF, where p ranges over P and j ranges over G: L(P, Sub) : ϕ ::= p (ϕ ϕ) (ϕ ϕ) (ϕ ϕ) j ϕ j ϕ (ϕ ϕ) (ϕ ϕ) (ϕ ϕ) The language L(P, Sub ) is the language L(P, Sub) without the (intuitionistic) connective. Definition 9 (Point set, accessibility relation). A point set P = (P, ) is a non-empty set P together with a partial order on P. The set P rop(p) of propositions on P is the set of all subsets X of P which are closed upwards: that is, if x X and x x then x X. When is the identity relation =, we say that P is flat. We abusively write x P for x P. A binary relation R is a positive two place accessibility relation on the point set P if, and only if, for any x, y P where xry, if x x then there is a y y such that x Ry. Similarly, if xry and y y then there is some x x such that x Ry. A binary relation R is a plump positive two-place accessibility relation on the point set P if, and only if, for any x, y, x, y P, where xry, x x and y y it follows that x Ry. A ternary relation R is a three place accessibility relation on the point set P if, and only if, whenever Rxyz and z z then there are y y and x x such that Rx y z. Similarly, if x x then there are y y and z z such that Rx y z, and if y y then there are x x and z z, such that Rx y z. A ternary relation R is a plump three-place accessibility relation on the point set P if, and only if, for any x, y, z, x, y, z P such that Rxyz, if x x, y y and z z, then Rx y z. We say that Q is an accessibility relation if, and only if, it is either a positive two-place or a three-place accessibility relation. Note that plump accessibility relations are accessibility relations. The definitions of accessibility relations relate R, R with. They are set in such a way that condition (Persistence) can be lifted to arbitrary formulas of L(P, Sub) and holds not only for the propositional letters of P. 9

11 Definition 10 (substructural model). A (multi-modal) substructural model is a tuple M = (P, R 1,... R m, R, I) where: P = (P, ) is a point set; R j P P is a (binary) accessibility relation on P, for each j G; R P P P is a (ternary) accessibility relation on P; I : P 2 P is a function called the interpretation function satisfying moreover the condition {x M x I(p)} P rop(p), which can be reformulated as follows: for all x, y P and all p P, if p I(x) and x y then p I(y). (Persistence) We abusively write x M for x P and (M, x) is called a pointed substructural model. The class of all pointed substructural models is denoted E. A (pointed) substructural frame is a (pointed) substructural model without interpretation function. The class of all pointed substructural frames is denoted F. Definition 11 (Evaluation relation). We define the evaluation relation E L(P, Sub) as follows. Let M be a substructural model, x M and ϕ, ψ L(P, Sub). The truth conditions for the atomic facts and the connectives of Sub are defined as follows: M, x always; M, x never; M, x p iff p I(x); M, x ϕ ψ iff M, x ϕ and M, x ψ; M, x ϕ ψ iff M, x ϕ or M, x ψ; M, x j ϕ iff for all y P, such that xr j y, M, y ϕ; M, x j ϕ iff there is y P such that yr jx and M, y ϕ; M, x ϕ ψ iff there are y, z P such that Ryzx, M, y ϕ and M, z ψ; M, x ϕ ψ iff for all y, z P such that Rxyz, if M, y ϕ then M, z ψ; M, x ψ ϕ iff for all y, z P such that Ryxz, if M, y ϕ then M, z ψ; M, x ϕ ψ iff for all y P, if x y then not M, y ϕ or M, y ψ. We extend these definitions to the class of pointed substructural frames. We define the evaluation relation F L(P, Sub) as follows. Let (F, x) be a pointed frame and let ϕ L(P, Sub). Then, we have that F, x ϕ iff for all interpretation functions I such that (F, I) satisfies Persistence, (F, I), x ϕ A substructural model stripped out from its interpretation function corresponds to a frame as defined in [86, Definition 11.8] and without truth sets. In [86], a model is a frame together with an evaluation relation. 10

12 Urquhart s semantics. The Urquhart s semantics for relevance logic was developed independently from the Routley Meyer s semantics in the early 1970 s. We present it because its configuration is even more directly related to the DEL framework than the Routley Meyer s semantics. An operational frame is a set of points P together with a function which gives us a new point from a pair of points: : P P P. (4) An operational model is then an operational frame together with a relation which indicates what formulas are true at what points. The truth conditions for the implication are defined as follows: x ϕ ψ iff for each y, if y ϕ then x y ψ (5) As one can easily notice, an operational frame is a Routley-Meyer frame where Rxyz holds if and only if x y = z. Hence, the ternary relation R of the Routley Meyer semantics is a generalization of the function of the Urquhart s semantics. Because it is a relation, it allows moreover to apply x to y and yield either a set of outcomes or no outcome at all. 3.2 Updates as Ternary Relations The ternary relation of the Routley and Meyer semantics was introduced originally for technical reasons: any 2-ary (n-ary) connective of a logical language can be given a semantics by resorting to a 3-ary (resp. n+1-ary) relation on worlds. Subsequently, a number of philosophical interpretations of this ternary relation have been proposed and we will briefly recall some of them at the end of this section (see [31, 87, 65] for more details). However, one has to admit that providing a non-circular and conceptually grounded interpretation of this relation remains problematic [31]. In this article, we propose a new dynamic interpretation of this relation, inspired by the ternary semantics of DEL. First, one should observe that the DEL product update of Definition 7 can be seen as a partial function F from a pair of pointed L P model and pointed L A model to another pointed L P model: F : E P E A E P (6) There is a formal similarity between this abstract definition of the DEL product update and the function of Expression (4) introduced by Urquhart in the early 1970s for providing a semantics to the implication of relevance logic. This similarity is not only formal but also intuitively meaningful. Indeed, the intuitive interpretation of the DEL product update operator is very similar to the intuitive interpretation of the function of Urquhart. Points are sometimes also called worlds, states, situations, set-ups, and as explained by Restall: We have a class of points (over which x and y vary), and a function which gives us new points from old. The point x y is supposed, on Urquhart s interpretation, to be the body of information given by combining x with y. [87, p. 363] 11

13 and also, keeping in mind the truth conditions for the connective of Expression (5): To be committed to A B is to be committed to B whenever we gain the information that A. To put it another way, a body of information warrants A B if and only if whenever you update that information with new information which warrants A, the resulting (perhaps new) body of information warrants B. (my emphasis) [87, p. 362] From these two quotes it is natural to interpret the DEL product update of Definition 7 as a specific kind of Urquhart s function (Expression (4)). Moreover, as explained by Restall, this substructural update can be nonmonotonic and may correspond to some sort of revision: [C]ombination is sometimes nonmonotonic in a natural sense. Sometimes when a body of information is combined with another body of information, some of the original body of information might be lost. This is simplest to see in the case motivating the failure of A B A. A body of information might tell us that A. However, when we combine it with something which tells us B, the resulting body of information might no longer warrant A (as A might with B). Combination might not simply result in the addition of information. It may well warrant its revision. (my emphasis) [87, p. 363] Our dynamic interpretation of the ternary relation is consistent with the above considerations: sometimes, updating beliefs amounts to revise beliefs. As it turns out, belief revision has also been extensively studied within the DEL framework and DEL has been extended to deal with this phenomenon [5, 109, 99, 25, 26, 61, 7]. More generally, an update can be seen as a partial function F from a pair of pointed L P model and pointed L A model to a set of pointed L P model: F : E P E A P(E P ) (7) Equivalently, an update can be seen as a ternary relation UL (C ϕ ) defined on E P E A between three pointed models ((M, w), (A, e), (M f, w f )) where (M, w) is a pointed L P model, (A, e) is a pointed L A model and (M f, w f ) is another pointed L P model: R E P E A E P (8) The ternary relation of Expression (8) then resembles the ternary relation of the Routley and Meyer semantics. This is not surprising since the Routley and Meyer semantics generalizes the Urquhart semantics (they are essentially the same, since as we explained it in the previous section, an operational frame is a Routley and Meyer frame where Rxyz holds if and only if x y = z). Viewed from the perspective of DEL, the ternary relation then represents a particular sort of update. With this interpretation in mind, R xyz reads as the occurrence of event y in world x results in the world z and the corresponding conditional χ ϕ reads as the occurrence in the current world of an event satisfying property χ results in a world satisfying ϕ. The dynamic reading of the ternary relation and its corresponding conditional is very much in line with the so-called Ramsey Test of conditional logic. The Ramsey test can be viewed as the 12

14 very first modern contribution to the logical study of conditionals and much of the contemporary work on conditional logic can be traced back to the famous footnote of Ramsey [83]. Roughly, it consists in defining a counterfactual conditional in terms of belief revision: an agent currently believes that ϕ would be true if ψ were true (i.e. ψ ϕ) if and only if he should believe ϕ after learning ψ. A first attempt to provide truth conditions for conditionals, based on Ramsey s ideas, was proposed by Stalnaker. He defined his semantics by means of selection functions over possible worlds f : W 2 W W. As one can easily notice, Stalnaker s selection functions could also be considered from a formal point of view as a special kind of ternary relation, since a relation R f W 2 W W can be canonically associated to each selection function f. Moreover, like the ternary relation corresponding to a product update (Expression (8)), this ternary relation is two-sorted : the antecedent of a conditional takes value in a set of worlds (instead of a single world). 5 So, the dynamic reading of the ternary semantics is consistent with the dynamic reading of conditionals proposed by Ramsey. This dynamic reading was not really considered and investigated by substructural logicians when they connected the substructural ternary semantics with conditional logic [31]. On the other hand, the dynamic reading of inferences has been stressed to a large extent by van Benthem [100, 103] (we will come back to this point in Section 7), and also by Baltag & Smets [23, 25, 26]. In particular, they distinguished dynamic belief revision from static (standard) belief revision: the latter is a revision of the agent s beliefs about the state of the world as it was before an event, and the former is a revision of the state of the world as it is after the event. Our dynamic interpretation of the ternary semantics of substructural logics is consistent with the interpretations proposed by substructural logicians. In fact, our point of view is also very much in line with the claim of Gärdenfors and Makinson [46, 63] that non-monotonic reasoning and belief revision are two sides of the same coin : as a matter of fact, non-monotonic reasoning is a reasoning style and belief revision is a sort of update. The formal connection in this case also relies on a similar idea based on the Ramsey test. To summarize our discussion, the DEL product update provides substructural logics with an intuitive and consistent interpretation of its ternary relation. This interpretation is consistent, in the sense that the intuitions underlying the definitions of the DEL framework are coherent with those underlying the ternary semantics of substructural logics, as witnessed by our quotes and citations from the substructural literature. Other interpretations of the ternary relation One interpretation, due to Barwise [28] and developed by Restall [85], takes worlds to be sites or channels, a site being possibly a channel and a channel being possibly a site. If x, y and z are sites, Rxyz reads as x is a channel between y and z. Hence, if ϕ ψ is true at channel x, it means that all sites y and z connected by channel x are such that if ϕ is information available in y, then ψ is information available in z (see the paragraph Related Proof Systems at the end of Section 5.3 for more details). Another similar interpretation due to Mares [64] adapts Israel and Perry s theory of information [78] to the relational semantics. In this interpretation, worlds are situations in the sense of Barwise and 5 Note that Burgess [34] already proposed a ternary semantics for conditionals, but his truth conditions and his interpretation of the ternary relation were quite different from ours. 13

15 Perry s situation semantics [27] and pieces of information called infons can carry information about other infons: an infon might carry the information that a red light on a mobile phone carries the information that the battery of the mobile phone is low. In this interpretation, the ternary relation R represents the informational links in situations: if there is an informational link in situation x that says that an infon σ carries the information that the infon π also holds, then if Rxyz holds and y contains the infon σ, then z contains the infon π. Other interpretations of the ternary relation have been proposed by Beall & Al. [31], with a particular focus on their relation to conditionality. For more information on this topic the reader is invited to consult [66] which covers the material briefly reviewed in this paragraph. 4 Update Logic In this section we define our update logic. After introducing some mathematical definitions in Section 4.1, we motivate in Section 4.2 the introduction of three triples of logical connectives. These connectives generalize the triple (,, ) of substructural logics and will be given a cyclical semantics in Section Preliminary Definitions The general definitions of this section will be used in the rest of the article. Definition 12 (Logic). A logic is a triple L := (L (P, C ϕ ), E, ) where L (P, C ϕ ) is a logical language defined as a set of well-formed expressions built from a set of logical (and structural) connectives C ϕ and a set of propositional letters P; E is a class of pointed models or frames; is a satisfaction relation which relates in a compositional manner elements of L (P, C ϕ ) to models of E by means of so-called truth conditions. Note that the above semantically based definition of a logic is also used by French et Al. [41]. Example 4. The triples (L(P, Sub), E, ) and (L(P, Sub ), E, ) are logics. We list in Figure 6 logics that we deem to be classical. The triple (L P, E P, ) is also a logic, called epistemic logic. Definition 13 (Expressiveness). Let two logics L = (L, E, ) and L = (L, E, ) be given (interpreted over the same class of models E). Let ϕ L and ϕ L. We say that ϕ is as expressive as ϕ when {M E M ϕ} = {M E M ϕ }. We say that L has at least the same expressive power as L, denoted L L, when for all ϕ L, there is ϕ L such that ϕ is as expressive as ϕ. When L has at least the same expressive power as L and vice versa, we say that L and L have the same expressive power and we write it L L. Otherwise, L is strictly more expressive than L and we write it L > L. Example 5. It holds that (L(P, Sub), E, ) > (L(P, Sub ), E, ). 14

16 Models E Connectives Cϕ Logic (L (P, C ϕ ), E, ) R R = Propositional Logic = j Modal Logic = Lambek Calculus = j Modal Lambek Calculus Figure 6: Classical Logics 4.2 Talking about Ternary Relations If we want to reason about updates, we must be able to express properties of updates. In other words, we need a language for talking about updates. Since we represent them by ternary relations, it seems natural to require that our language be able to express properties that relate what is true at each point of the ternary relations, that is, what is true at: 1. the initial situation (expressed by a formula ϕ), 2. the event occurring in this situation (expressed by a second formula χ), 3. the resulting situation after the event has occurred (expressed by a third formula ψ): 1, ϕ 2, χ 3, ψ This leads us to the following general question: assume that we stand in one of these three time points x (be it 1, 2 or 3), what kind of property can we express and infer about the other time points y and z? Here is a non-exhaustive list of the possible and most natural expressions that we would want to state: (a) For all y, if y satisfies ϕ then for all z, z satisfies ψ: x y z(ϕ(y) ψ(z)). For example, in the initial state 1, is it the case that any event satisfying χ will always lead to a state 3 satisfying ψ? Or, in state 3, is it the case that before the occurrence of any event satisfying χ, ϕ held in all initial states 1? (b) There exist y and z such that y satisfies ϕ and z satisfies ψ: x y z(ϕ(y) ψ(z)). For example, in state 1, is it the case that there exists an event satisfying χ that may lead to a state where ψ holds? Or, in state 3, is it possible that our current state might have been the result of an event satisfying χ in an initial state where ϕ held? (c) For all y satisfying ϕ, there exists z satisfying ψ: x y z(ϕ(y) ψ(z)). For example, in state 1, is it the case that any events satisfying χ may lead possibly to a state where ψ holds? Or, in state 3, is it the case that an event satisfying χ might have occurred so that any former situation before this event satisfied ϕ? This list of expressions is obviously non-exhaustive. Providing formal tools that answer these kinds of questions leads to applications in artificial intelligence and theoretical computer 15

17 science, and as it turns out, some of these questions have already been addressed in DEL and other logical formalisms (see Section 7 for more details and examples). Typically, most of the works about conditionals and belief dynamics deal with the first kind of statements (a) or (b). In fact, the conditionals and of substructural and relevance logics of the previous section are of the form (a), whereas the substructural connective is of the form (b). The language that we will define will only deal with the first two kinds of expressions (a) and (b) (Section 4.3). This language is intended to capture the various conditionals and belief change operators which have been introduced in the philosophical and artificial intelligence literature. 4.3 Syntax and Semantics of Update Logic We define formally formulas, structures and then consecutions (sometimes called sequents in the literature). This is an incremental definition and each of these objects is defined on the basis of the previous one. Moreover, in the sequel, we will view sets of formulas, sets of structures and sets of consecutions as logical languages. Notation 2. In the rest of this article, we will use the following logical connectives Con ϕ and structural connectives Con X (ϕ will denote formulas and X will denote structures): { } Con ϕ := j, j,,,,,, i, i, i i {1, 2, 3}, j G Con X := {, j,, i i {0, 1, 2, 3}, j G} The connectives,, i, i, i,, 0,, i (where i ranges over {1, 2, 3}) are binary connectives, j, j,,, j are unary connectives (where j ranges over G) and, are nullary connectives. The structural connective, 0 will often simply be denoted,. Definition 14 (Formula, structure and consecution). Let C ϕ Con ϕ be a non-empty set of logical connectives. The language associated to C ϕ, denoted L (P, C ϕ ), is the language built compositionally from the connectives of C ϕ and the set of propositional letters P. Elements of the language L (P, C ϕ ) are called L (P, C ϕ ) formulas and are generally denoted ϕ, χ, ψ,... Let C X Con X and C ϕ Con ϕ be non-empty sets of structural connectives and logical connectives. The set of structures associated to C ϕ and C X, denoted S (P, C ϕ, C X ), is the language built compositionally from the structural connectives of C X and the set L (P, C ϕ ). Elements of the language S (P, C ϕ, C X ) are called S (P, C ϕ, C X ) structures and are generally denoted X, Y, Z,... The structural connectives associated to C ϕ, denoted Struc(C ϕ ), is the set of structural connectives {,, 0 } together with {, i i } {1, 2, 3}} if C ϕ { i, i, i i {1, 2, 3}} and with { j } if C ϕ { j, j j G. We denote by S (P, C ϕ ) the set of all S (P, C ϕ, Struc(C ϕ )) structures. Let C X Con X and C ϕ Con ϕ be non-empty sets of structural connectives and logical connectives. A S (P, C ϕ, C X ) consecution is an expression of the form X Y, 16

18 X or Y, where X, Y S (P, C ϕ ). The S (P, C ϕ, C X ) structure X is called the antecedent and the S (P, C ϕ, C X ) structure Y is called the consequent. Moreover, we write X 1,... X n Y for ((... (X 1,..., X n 2 ), X n 1 ), X n ) Y. We denote by C(C ϕ ) the set of all S (P, C ϕ, Struc(C ϕ )) consecutions. The connective (standing for the material implication) is defined by the following abbreviation: (ϕ ψ) := ( ϕ ψ). To avoid any ambiguity, every occurence of any binary connective is surrounded by brackets. Example 6. If C ϕ = {,, 3, 1, 2 }, then the language L (P, C ϕ ) is defined by the following grammar in BNF, where p ranges over P: L (P, C ϕ ) : ϕ ::= p ϕ (ϕ ϕ) (ϕ 3 ϕ) (ϕ 1 ϕ) (ϕ 2 ϕ) Then, we have that Struc(C ϕ ) = {,, 0,, i i {1, 2, 3}}. So, the language S (P, C ϕ ) := S (P, C ϕ, Struc(C ϕ )) is defined by the following grammar in BNF, where ϕ ranges over L (P, C ϕ ) and i ranges over {1, 2, 3}: S (P, C ϕ ) : X ::= ϕ X (X, 0 X) (X, i X) Notation 3. To save parenthesis, we use the following ranking of binding strength: i, i, i,,, (where i ranges over {1, 2, 3}). For example, 1 p q r 3 s stands for (( 1 ( p)) q) (( r) 3 s) (additional brackets have been added for the unary connectives 1 and, even if they are not needed and will not appear in any formula anyway). For every binary connective, we use the following notation: X 1... X n := ((... (X 1... X n 2 ) X n 1 ) X n ). For example, ϕ 1... ϕ n := ((... (ϕ 1... ϕ n 2 ) ϕ n 1 ) ϕ n ) and X 1,..., X n := ((... (X 1,..., X n 2 ), X n 1 ), X n ). Definition 15 (Update logic). Let E be an arbitrary set of three elements. For each i {1, 2, 3}, we define the cyclical permutations σ i : E 3 E 3 as follows: for all x, y, z E, σ 1 (x, y, z) = (x, y, z) σ 2 (x, y, z) = (z, x, y) σ 3 (x, y, z) = (y, z, x). We define the evaluation relation E L (P, Con ϕ ) inductively as follows. Let (M, x) E be a pointed substructural model and let ϕ L (P, Con ϕ ). The truth conditions for the connectives j, j,,,, are defined like in Definition 11. The truth condition for the Boolean negation is defined as follows: M, x ϕ iff it is not the case that M, x ϕ. The truth conditions for the connectives i, i, i are defined as follows: for all i {1, 2, 3}, we have that M, x ϕ i ψ iff there are y, z P such that σ i (x, y, z) R, M, y ϕ and M, z ψ; M, x ϕ i ψ iff for all y, z P such that σ i (x, y, z) R, if M, y ϕ then M, z ψ; M, x ϕ i ψ iff for all y, z P such that σ i (x, y, z) R, if M, z ψ then M, y ϕ. 17

19 We extend the scope of the evaluation relation simultaneously in two different ways in order to also relate points to S (P, Con ϕ ) structures. The antecedent evaluation relation A E S (P, Con ϕ ) is defined inductively as follows: for all i {1, 2, 3}, M, x A ϕ iff M, x ϕ; M, x A X iff it is not the case that M, x K X; M, x A j X iff there is y M such that yr j x and it holds that M, y A X; M, x A X, Y iff M, x A X and M, x A Y ; M, x A X, i Y iff there are y, z M such that σ i (x, y, z) R, M, y A X and M, z A Y. The consequent evaluation relation follows: for all i {1, 2, 3}, K E S (P, Con ϕ ) is defined inductively as M, x K ϕ iff M, x ϕ; M, x K X iff it is not the case that M, x A X; M, x K j X iff for all y M such that xr j y, it holds that M, y K X; M, x K X, Y iff M, x K X or M, x K Y ; M, x K X, i Y iff for all y, z M such that σ i (x, y, z) R, M, y K X or M, z K Y. We extend the scope of the relation to also relate points to S (P, Con ϕ ) consecutions. Depending on the form of the S (P, Con ϕ ) consecution, that is, whether it is of the form X Y, Y or X, we have: M, x X Y iff if M, x A X, then M, x K Y ; M, x Y iff M, x K Y ; M, x X iff it is not the case that M, x A X. So, for all C ϕ Con ϕ, the triples (L (P, C ϕ ), E, ), (S (P, C ϕ ), E, A ), (S (P, C ϕ ), E, K ) and (C(C ϕ ), E, ) are logics (as defined in Definition 12). The triple (L (P, Con ϕ ), E, ) is also a logic, called update logic. Spelling out the truth conditions for the connectives i, i and i for i {1, 2, 3}, we obtain the expressions of Figure 7. The indices 1, 2 and 3 of our connectives indicate when formulas are evaluated. The connectives 1, 1 and 1 express properties of updates before the event, the connectives 2, 2 and 2 properties during the event and the connectives 3, 3 and 3 properties after the event. Typically, the formula ϕ deals with the initial situation, the formula χ deals with the event and the formula ψ deals with the final situation. The direction of the arrow ( or ) indicates the conditional direction in which the formula should be read. For example, the formula ψ 2 ϕ tells us that it should be evaluated during an event (2) and reads as if the final situation will satisfy ψ then the initial situation must necessarily satisfy ϕ, whereas ψ 2 ϕ reads as if the initial situation satisfies ϕ then the final situation will 18

20 M, x χ 1 ψ iff there are y, z P such that Rxyz, M, y χ and M, z ψ; M, x χ 1 ψ iff for all y, z P such that Rxyz, if M, y χ then M, z ψ; M, x χ 1 ψ iff for all y, z P such that Rxyz, if M, z ψ then M, y χ; M, x ψ 2 ϕ iff there are y, z P such that Rzxy, M, z ϕ and M, y ψ; M, x ψ 2 ϕ iff for all y, z P such that Rzxy, if M, y ψ then M, z ϕ; M, x ψ 2 ϕ iff for all y, z P such that Rzxy, if M, z ϕ then M, y ψ; M, x ϕ 3 χ iff there are y, z P such that Ryzx, M, y ϕ and M, z χ; M, x ϕ 3 χ iff for all y, z P such that Ryzx, if M, y ϕ then M, z χ; M, x ϕ 3 χ iff for all y, z P such that Ryzx, if M, z χ then M, y ϕ. Figure 7: Spelling out the Truth Conditions necessarily satisfy ψ. The formula χ 1 ψ reads as ψ will hold after the occurrence of any events satisfying χ and the formula ϕ 3 χ reads as ϕ held before the occurrence of any events satisfying χ. The connectives 1, 2, 3 are of the form (b) and the connectives 1, 1, 2, 2, 3, 3 are of the form (a) (see page 15). Note that the classical substructural connectives, and of the previous section correspond to our connectives 3, 1 and 2. So, our language L (P, Con ϕ ) extends the language L(P, Sub ) of substructural logics presented in Section 3.1 and the logic (L (P, Con ϕ ), E, ) is therefore at least as expressive as (L(P, Sub ), E, ). In fact, (L (P, Con ϕ ), E, ) is strictly more expressive than (L(P, Sub ), E, ), as proved in [12]. 5 Proof calculi for Update Logic Extending Gentzen s original sequent calculi with modalities has turned out over the years to be difficult. Many of the interesting theoretical properties of sequent calculi are lost when one adds modalities (see for example Poggiolesi [80, Chapter 1] for more details). A number of methods have been proposed to overcome these difficulties: display calculi, labelled sequents, tree hypersequents (see Poggiolesi and Restall [81] for an accessible introduction to these different sorts of calculi). In this section, we provide a display calculus and a sequent calculus for our update logic. As we will see, this display calculus will be a generalization of the display calculus for modal logic introduced by Wansing [116] and the sequent calculus will be a generalization of 19

21 the non-associative Lambek calculus NL [59, 60]. 5.1 Preliminary Definitions The general definitions of this section will be used in the rest of the article. Definition 16 (Truth, validity, logical consequence). Let L = (L, E, ) be a logic. Let M E, Γ L and ϕ L. We write M Γ when for all ψ Γ, we have M ψ. Then, we say that ϕ is true (satisfied) at M or M is a model of ϕ when M ϕ; ϕ is a logical consequence of Γ, denoted Γ Lϕ, when for all M E, if M Γ then M ϕ; ϕ is valid, denoted Lϕ, when for all models M E, we have M ϕ. Definition 17 (Conservativity). Let L = (L, E, ) and L = (L, E, ) be two logics such that L L. We say that L is a conservative extension of L when { ϕ L Lϕ } = L { ϕ L L ϕ }. Our definition of a proof system and of an inference rule is taken from Mendelson [67]. Definition 18 (Proof system and sequent calculus). Let L = (L, E, ) be a logic. A proof system P for L is a set of elements of L called axioms and a set of inference rules. Most often, one can effectively decide whether a given element of L is an axiom. To be more precise, an inference rule R in L is a relation among elements of L such that there is a unique l N such that, for all ϕ, ϕ 1,..., ϕ l L, one can effectively decide whether (ϕ 1,..., ϕ l, ϕ) R. The elements ϕ 1,..., ϕ l are called the premises and ϕ is called the conclusion and we say that ϕ is a direct consequence of ϕ 1,..., ϕ l by virtue of R. Let Γ L and let ϕ L. We say that ϕ is provable (from Γ) in P or a theorem of P, denoted P ϕ (resp. Γ P ϕ), when there is a proof of ϕ (from Γ) in P, that is, a finite sequence of formulas ending in ϕ such that each of these formulas is: 1. either an instance of an axiom of P (or a formula of Γ); 2. or the direct consequence of preceding formulas by virtue of an inference rule R. If S is a set of L consecutions, this set S can be viewed as a logical language. Then, we call sequent calculus for S a proof system for S. Definition 19 (Soundness and completeness). Let L = (L, E, system for L. Then, ) be a logic. Let P be a proof P is sound for the logic L when for all ϕ L, if P ϕ, then P is (strongly) complete for the logic L when for all ϕ L (and all Γ L), if Lϕ, then P ϕ (resp. if Γ L ϕ, then Γ P ϕ). Because a proof is a finite sequence of formulas, soundness for the logic L coincides with strong soundness for the logic L, i.e. for all ϕ L and all Γ L, if Γ P ϕ, then Γ L ϕ. Lϕ. 20